A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First- and second-order optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and a time-independent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerical example is provided.

References:

[1]

S. Aniţa, V. Arnăutu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011. From mathematical models to numerical simulation with MATLAB®.
doi: 10.1007/978-0-8176-8098-5.

T. Bayen and F. J. Silva,
Second order analysis for strong solutions in the optimal control of parabolic equations, SIAM Journal on Control and Optimization, 54 (2016), 819-844.
doi: 10.1137/141000415.

F. J. Silva,
Second order analysis for the optimal control of parabolic equations under control and final state constraints, Set-Valued and Variational Analysis, 24 (2016), 57-81.
doi: 10.1007/s11228-015-0337-4.

[29]

F. Tröltzsch, Optimal Control of Partial Differential Equations, volume 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels.
doi: 10.1090/gsm/112.

show all references

References:

[1]

S. Aniţa, V. Arnăutu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011. From mathematical models to numerical simulation with MATLAB®.
doi: 10.1007/978-0-8176-8098-5.

T. Bayen and F. J. Silva,
Second order analysis for strong solutions in the optimal control of parabolic equations, SIAM Journal on Control and Optimization, 54 (2016), 819-844.
doi: 10.1137/141000415.

F. J. Silva,
Second order analysis for the optimal control of parabolic equations under control and final state constraints, Set-Valued and Variational Analysis, 24 (2016), 57-81.
doi: 10.1007/s11228-015-0337-4.

[29]

F. Tröltzsch, Optimal Control of Partial Differential Equations, volume 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels.
doi: 10.1090/gsm/112.

Figure 1.
Values of the state $y_1$ and the control $u_1$, for different values of the time.